On a class of new hypergeometric transformations

In this article we continue the investigations presented in our previous papers [1,2,3,4], presenting some, for the best of our knowledge, new transformations of the Gauss hypergeometric function (3) and (13). They have been obtained using only elementary methods and stem from a couple of integrals evaluated in terms of complete elliptic integral of
first kind by Legendre in [5] Chapter XXVII, at sections II and III.